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%I #14 Jan 24 2022 20:51:48
%S 0,10,24,26,39,3573,26899,68697,497699,3559595,555959597395
%N Numbers for which iteration of the powertrain map of A133500 takes a record number of steps to converge.
%C Where records occur in A133501.
%C This sequence is almost certainly finite.
%C The number 31395559595973 takes 16 steps to converge and may be the next term. It may also be the last term.
%C The next term is > 10^7 (and <= 31395559595973).
%e The smallest number that takes 13 steps to converge is 497699, for which the trajectory is 497699 -> 11948427342082473984 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
%e The smallest number that takes 15 steps to converge is 3559595 -> for which the trajectory is 3559595 -> 4634857177734375 -> 23122964691361341376561152 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
%e The number 31395559595973 takes 16 steps to converge and so the next term is >= 16. The trajectory is 31395559595973 -> 471570692025125026702880859375 -> 34755118508614725279865110528 -> 23122964691361341376561152000000 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
%e The smallest number that takes 16 steps to converge is 555959597395, for which the trajectory starts 555959597395 -> 471570692025125026702880859375 and then continues as above. - _Michael S. Branicky_, Jan 24 2022
%Y See A133508 for the corresponding numbers of steps. Cf. A133500, A133501.
%Y See also A003001.
%K nonn,base
%O 1,2
%A _J. H. Conway_ and _N. J. A. Sloane_, Dec 03 2007, Dec 04 2007, Dec 18 2007
%E a(11) from _Michael S. Branicky_, Jan 24 2022